J Solid State Electrochem (2013) 17:3075–3081 DOI 10.1007/s10008-013-2158-4
ORIGINAL PAPER
A systematic approach to the impedance of surface layers with mixed conductivity forming on electrodes László Péter
Received: 30 April 2013 / Revised: 10 June 2013 / Accepted: 12 June 2013 / Published online: 30 June 2013 # Springer-Verlag Berlin Heidelberg 2013
Abstract Electrode impedance can be evaluated on the basis of the electrode reaction kinetics in many systems, even for complicated electrode reactions. However, when a surface layer is present on the electrode surface, the theoretically well-established impedance model of the electrode reaction is often completed with phenomenological equivalent circuit elements in order to achieve the number of time constants as derived from the electrode impedance spectra measured. In these cases, the meaning of the phenomenological equivalent circuit elements are often unclear, though the presence of these elements is helpful to describe the system throughout the frequency domain used for the measurement. In the present work, an attempt will be shown to separate the effect of the electronic and ionic charge transfer in a surface layer and to identify the appropriate equivalent circuits. Examples are shown from the fields of lithium-ion batteries where a solid electrolyte interface as a surface layer is present at the negative electrode and the contribution of various charge carriers may be of importance. Keywords Electrode impedance . Equivalent circuit . Solid electrolyte interface (SEI) . electronic and ionic conductivity
Introduction Electrochemical impedance spectroscopy (EIS) is a wellestablished technique to reveal mechanistic details of various electrode processes. EIS is a popular and convenient tool to identify the contribution of the charge transfer, the diffusion of the reactant, the adsorption of the reactants and/or the intermediates, and the intercalation of the ions. The abovementioned steps – or those that are indeed a part of L. Péter (*) Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O. Box 49, Budapest 1525, Hungary e-mail:
[email protected]
the actual electrode process – together make the impedance of the electrode reaction (ZER). The impedance or admittance functions of many electrode reactions can be calculated with exact mathematical methods, even though it might not be possible to construct an equivalent circuit in all cases. Besides the electrode reaction, the impedance measured for a particular electrode as a whole comprises additional circuit elements. Due to the electric double layer that builds up between the first-order and second-order conducting phases (that are mostly a metal and an electrolyte solution, respectively), the interface exhibits an inherent capacitive nature. This is represented in the electrode impedance with the so-called double layer capacitance (CDL), which is connected parallel to the impedance of the electrode reaction, indicating the parallel ways of the current flow. A serial resistance element (that is called in most cases the solution resistance) must also be taken into account to evaluate the data measured. This element is unrelated to the electrode processes themselves but accounts for the cell geometry; namely, the non-infinitely close position of the potential reference point to the working electrode certainly leads to an ohmic drop. The solution resistance is a serial circuit element. The conventional representation of the equivalent circuit of an electrode reaction can be depicted with the circuit as shown in Fig. 1a. When the impedance of the electrode reaction includes the charge transfer resistance and diffusion (Warburg) impedance, it is known as the Randles circuit. However, the above-described treatment of the electrode impedance can be applied under special circumstances only. The most important criterion is that the electrolyte components accounting for the double layer behavior have to be present in a large enough concentration (supporting electrolyte). Also, it should be provided that the double layer charge/discharge process is fully independent of both the electrode reaction and the specific adsorption on the electrode surface, i.e., the supporting electrolyte must be electrochemically inactive. If this condition is not fulfilled, the equivalent circuit of the electrode process needs to be
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a
surface layer can have a resistive contribution to the total electrode impedance. (2) On the negative (carbonaceous or Li) electrode of Liion batteries, an intermediate layer must form to provide the functionality of the electrode and to make it possible to intercalate Li in a reversible manner. This layer is generally called the solid electrolyte interface (SEI) that is created by the first charge–discharge cycle of the cell and which also protects the electrolyte solution from any further decomposition.
CDL
RS
ZER
b CDL RS CSL
ZER
RSL
c RS
CSL RSL
CDL ZER
Fig. 1 a The generally accepted equivalent circuit for an electrode reaction where no additional surface layer is present on the electrode surface. b and c Frequently applied equivalent circuits for modeling electrode impedance where a solid surface layer is present on the electrode surface. SL denotes the surface layer, and the other notations are the same as in the main text
modified, and even a diffusion impedance element may have to be included if the species responsible for the capacitive processes are present in a small concentration only [1, 2]. In the case when the species taking part in the Faraday reaction are the same as those responsible for the double layer properties, the Warburg impedance is often connected in series with the (CDL║RCT) branch. This is often the case when electrodes for lithium-ion batteries are studied, and the fit of the parameters using such circuits are generally satisfactory [3–10]. Nevertheless, such circuits have not been theoretically established so far. All features mentioned above hold for an electrode in which there is a direct contact between the electronconducting and the single ion-conducting phases. This is a scheme in which the interface with capacitive properties can be unambiguously identified, and the resistance RS can be connected serial with all other circuit elements. However, in many kinds of electrode, the electron-conducting phase is covered with another solid layer and, hence, the interface where the Faraday reaction takes place is separated from the electrolyte solution. Two typical examples are shortly described below: (1) When a corroding electrode is covered with a passive (and typically oxide type) surface layer, the oxidation of the metal takes place at the metal/passive layer interface. The oxide layer may exhibit a mixed conductivity, and all possible interfaces may have their capacitive contribution to the overall electrode behavior. Similarly, the
The conduction mechanism of the surface layers on the electrodes is difficult to describe since they cannot be usually synthesized in bulk form. Nevertheless, the presence of an additional non-insulating solid layer at the electrode surface is known to increase the number of time constants in the alternating current behavior of the electrode. Due to the insufficient knowledge on the conduction mechanism and the uncertainty in the involvement of such layers in the electrode process, a phenomenological approach is usually applied to account for the behavior of these layers. Namely, a parallel RC subcircuit is added to the theoretically established impedance related to the electrode reaction. The possible ways of adding this sub-circuit in order to account for the number of the time constants found experimentally are shown in Fig. 1b, c. Apparently, the circuit shown in Fig. 1b is widespread [11–13] but not exclusively used [14, 15] in the corrosion literature, while researchers dealing with lithium-ion batteries favor the circuit shown in Fig. 1c [16–34] in an overwhelming majority. Another concept is applied by Lewandowski and coworkers [35, 36] who prefer an REL −ZW −{CSEI‖(RSEI(CDL‖RCT))} type equivalent circuit, which exhibits time constants in the same number as the circuits shown in Fig. 1b, c. It is difficult to judge the quality of the models since practically, no paper reports on the impedance data fitted with more than a single model circuit; therefore, there is a great room for elucidation trials. Although the quantitative description of the alternating current behavior of the coated electrodes becomes satisfactory with the inclusion of one or even several [37–44] additional parallel RC sub-circuits, this may also give rise to some doubts concerning the relevance of the phenomenological approach: (1) No processes are identified that give rise to the occurrence of an additional time constant in the overall electrode behavior. Therefore, the fit values of the circuit elements added may miss any meaning or, at least, these parameters cannot be connected to elemental processes. (2) It is doubtful with which other circuit elements the double layer capacitance is to be connected in parallel.
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Depending on the assumption on the location of the capacitive interface, the resistance of the surface layer must be taken into account, in connection with either the ionic or electrical transport in this layer. In the following part, an attempt will be made to take into account the role of the surface layer in a theoretically correct manner.
Theoretical Below, an electrode with a solid surface layer will be considered. In the surface layer, both ionic and electron transport will be allowed. The charge transfer caused by the ion motion (diffusion or migration) and the electron transport will be considered as independent and parallel conduction channels. In this model, two interfaces appear: one between the electrode metal and the solid surface layer and another between the surface layer and the electrolyte solution. Both interfaces are allowed to serve as a capacitive one. Therefore, the inner part of the surface layer may behave in a similar way as the electrolyte solution in a simple electrode, i.e., the charge accumulated at the metal surface is compensated by a diffuse ion cloud in the solid surface layer. At the same time, the outmost boundary of the solid surface layer is also able to accumulate some extra charge, the compensation of which is provided by the ion redistribution in the Helmholtz layer of the solution, similarly to the conventional solid electrodes. The charge transfer to both surfaces with capacitive properties is hindered by a respective resistance originating from either the ion or the electron transfer through the solid surface layer. The charge transfer between chemical species (i.e., the electrode reaction itself) is allowed to take place at a single boundary only. Hence, all possible kinds of electrochemical reactions occur at the
Fig 2 The suggested equivalent circuit of an electrode covered with a solid surface layer exhibiting mixed electronic and ionic conductivity. Dashed lines indicate the phase boundaries, and the thick dotted line separates the parts of the system where electrical and ionic means of charge transfer are dominant
boundary of the first-order conductor/SEI interface only. The schematic representation of the equivalent circuit of this layer structure is depicted in Fig. 2 showing also both the geometric separation of the layers and the independent conduction channels. The immediate consequence of the separation of the roles of the charge carriers is that the high-frequency limit of the real part of the impedance will be RSOL +(RSL,e–1 +RSL,ion–1)–1. Therefore, the high-frequency limit has a different meaning as compared to the case of a bare metal electrode when the resistance of the surface layer is substantial. Depending on the density and on the mobility of the charge carriers in the surface layer, either the electrical or ionic resistance of the surface layer may have a dominant contribution to the highfrequency impedance of the system. In any case, the highfrequency impedance is no longer a simple and meaningless parameter to be deduced from the total impedance. It has to be noticed that the equivalent circuit shown in Fig. 2 necessarily leads to a so-called depressed semicircle behavior (referring to the Nyquist plot of the electrode impedance). As the frequency of the measurement decreases, the resistance in series to the impedance of the electrode process will become RSOL +RSL,ion, which is higher than the high-frequency limit of the real part of the impedance. Therefore, the high-to-medium frequency part of the impedance is stretched along the real axis of the Nyquist plot, giving rise to a depressed semicircle. Although such depressed semicircles are often modeled with a constant phase element (CPE) replacing a capacitor in a regular equivalent circuit, the present approach draws the attention to the fact that the application of the CPE is often a phenomenological tool to describe a system quantitatively when our knowledge on the system is insufficient. Although the complete quantitative analysis of the circuit shown in Fig. 2 is difficult, some trends can be clearly established. The present model can also be responsible for the occurrence of a new high-frequency semicircle in the
electron transport solution
solid surface layer
CSL/SOL
electrode metal
RSL,e RSOL
CME/SL RSL,ion ZER ionic transport
electric lead
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capacitors can be taken as elements with infinitely high impedance. The occurrence of the semicircle related to the capacitance of the solution side of the surface layer is possible until the total impedance related to the electron transfer through the surface layer is lower or only comparably high to the other resistive elements. The separation of the two arcs in the Nyquist plot is related to the difference in the time constants of the processes related to the electron and ion transfer through the surface layer. Figure 3c indicates that once the ionic conductivity of the surface layer is reduced, the entire impedance diagram shifts along the real axis. Although RSL in the diagrams shown in
Nyquist plot, similarly to the circuits shown in Fig. 1, c. The two semicircles are in some cases well expressed, while the formation of a single depressed semicircle is also possible. A few simulated results are shown in Fig. 3 where the impedance of the electrode reaction was taken to be a simple resistor (RER) for the sake of simplicity. It can be seen that, apart from extreme cases, the properties of the solid surface layer influence the high-frequency behavior of the system but lead to the same low-frequency behavior as for any other combination of the circuit elements related to the surface layer. At sufficiently low frequencies where the diffusion process can be noticed as the occurrence of a Warburg-type impedance, the circuit in Fig. 2 can be simplified and
-Im(Z ) / ohm cm
2
R SL,e / ohm cm2 =
10
a
1 4 10 25 100
1 kHz
10 Hz 100 Hz
5
0 0
5
10
15
20
Re(Z ) / ohm cm
2
-Im(Z ) / ohm cm
25
b
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0.08 0.2 0.6 2 9
10
30
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C ME/SL / mF cm-2 =
1 Hz
5
1 kHz
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0 0
5
10
15
20
Re(Z ) / ohm cm
2
25
30
2
c
R SL,ion / ohm cm2 =
12
-Im(Z ) / ohm cm
Fig. 3 Typical impedance diagrams obtained for the circuit shown in Fig. 2 as a function of some parameters. RSOL =0.5Ω cm2 and CSL/SOL =2×10−5 F cm−2 for all curves; dashed lines indicate the frequency of the perturbation. a CME/SL =6×10−4 F cm−2, ZER =RER =15Ω cm2, RSL,ion =15Ω cm2; RSL,e is indicated in the legend. b RSL,e =1Ω cm2, ZER =RER =15Ω cm2, RSL,ion =15Ω cm2; CME/SL is indicated in the legend. c CME/SL =6×10−4 F cm−2, ZER =RER =19.5Ω cm2; RSL,e =4Ω cm2, RSL,ion is indicated in the legend
10 Hz
1 4 6 8
10 8 6 4
100 Hz
2
1 Hz
0 0
5
10
15
Re(Z ) / ohm cm
20 2
25
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b
35
1
Phase angle / 0.5 PI
30
Z / ohm cm
-2
25 20 15 10
0.8
0.6
0.4
0.2
5 0
0
0.1
10
1000
100000
0.1
lg ( f / Hz)
10
1000
100000
lg ( f / Hz)
Fig. 4 Comparison of two impedance functions calculated with different equivalent circuits (Bode plot). The symbols indicate the equivalent circuit as shown in Fig. 1c using the following parameters: RS =1.452Ω cm2, RSL =17.908Ω cm2, CSL =0.185 mF cm−2, ZER =RER =11.14Ω cm2, and
CDL =0.0252 mF cm−2. The line indicates the equivalent circuit shown in Fig. 2 using the same parameters as in Fig. 3b, CME/SL =0.2 mF cm−2 (see the open squares in Fig. 3b). a: total impedance, b: phase angle of the impedance
Fig. 1b, c is introduced pronouncedly for the resistance of the surface layer, its change does not modify the high-frequency behavior, unlike the parameter RSL,ion in the circuit presented in Fig. 2. Figure 4 shows the comparison of two impedance functions calculated with different equivalent circuits. Although the calculation of the exact equivalence conditions is not easy, it can be seen that the impedance functions calculated with different circuits correspond to each other at least at the same level that is accepted during the fit of experimental impedance functions.
worse when a constant phase element is needed for the description of the electrode behavior [4, 6–8, 10, 17, 18, 22, 26, 29–34, 45, 46]. No wonder that, in many studies, only the resistive elements obtained from the fit of the experimental impedance data are given [3, 6–9, 16, 21, 23–25, 28–30, 32–34, 36, 37, 43, 44, 46, 47], and the indication of all parameters fitted is rather scarce [18, 26, 31, 38, 40, 45]. The impedance of an electrode applied in lithium-ion batteries can seldom be fitted with the classical Randles circuit [48, 49]. It is a common experience of the studies performed for both the negative electrodes of the Li-ion batteries and the entire cells themselves that an increase of the high-frequency impedance is obtained upon cycling the system [10, 39–41, 45–47, 49–55]. Whichever popular model is used to describe this behavior (see e.g., Fig. 1b, c), the high-frequency impedance should be invariant, unless something happens to the electrolyte. However, a large electrolyte excess is customarily used for the study of single electrodes, and the solute content of a full Li-ion cell is presumably unchanged. Therefore, the positive shift of the impedance functions along the real axis of the Nyquist plot cannot be explained on the basis of the conventional models. The model proposed in this work and shown as the equivalent circuit in Fig. 2 offers a natural explanation for the rise of the high-frequency impedance because a conductivity decrease in SEI due to aging leads to the enhancement of the real part of the impedance and does not require to include any capacitive element with an ill-defined meaning. This is well exemplified by the impedance diagrams shown in Fig. 3c.
Comparison with result for electrode covered with a solid electrolyte interface SEI formation on the negative electrode of a lithium-ion battery is an important but badly understood process. On the surface of both carbon-type and metallic Li negative electrodes, the decomposition of the solvent results in an insoluble and presumably polymer-type layer that plays a crucial role in the functionality of the electrode. The formation of SEI can be followed by the excess charge needed for the first charging process. Due to the lack of an ex-situ synthesis method, the properties of SEI cannot be studied independently of the electrode itself. The big majority of the impedance studies of the negative electrodes of Li batteries underpin that the number of time constants is larger than what is needed for the description of the metal deposition/dissolution (for Li anode) or intercalation/de-intercalation (for carbon-type anode). Therefore, an (CSL║RSL) sub-circuit is added to the system for the mathematical description of the alternating current behavior. Although it is straightforward to elucidate the meaning of the resistive element added, the explanation of the capacitive element is rather troublesome. The complications are even
Conclusion The impedance of an electrode covered with a surface layer of mixed conductivity was considered. It was suggested that the electronic and ionic currents passed through the surface
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layer can be considered as parallel conduction channels. Sample impedance diagrams were shown that are in agreement with the impedance functions normally obtained for electrodes covered with a surface layer. In the model proposed, the capacitances of the inner and outmost boundary of the surface layer are naturally separated, and there is no need to connect an (C║R) sub-circuit to either of the arm of the impedance function, eliminating the difficulty of the explanation of the meaning of the capacitance in this part of the equivalent circuit. It was also noted that the blind correction of the “solution resistance” may lead to the loss of important data that are indicative of the behavior of the system studied. An overview of the literature of the solid electrolyte interface showed that the model proposed is worthwhile of testing for electrode impedance measurements, and the change in the high-frequency limit of the impedance data measured may gain a natural explanation. Acknowledgement The author gratefully acknowledges Dr. Allen J. Bard for the postdoctoral research period spent in his lab in 1995–1997. This work was supported by the Hungarian Scientific Research Fund (OTKA) through grant # K 104 696.
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